# Statistical tests for paired or matched data

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(This is a sub-page of the Medical statistics page.)

## Paired t-test

Uses paired or matched data - same subjects before and after intervention, for example.

### Assumptions for paired t-test:

• independent observations;
• interval scale, or ordinal scale with many alternatives.
• Normal Distribution(s);
• no skew.

### Method for paired t-test:

For each pair, find difference. Calculate mean and standard deviation of the differences.

n = number of pairs of observations
m = mean of differences
s = standard deviation of the mean of differences
CR = m/(s/√n)
No of degrees of freedom = n-1.

In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic

$t = \frac{m - \mu_0}{s / \sqrt{n}},$

The constant μ0 is usually zero, but would be non-zero if you want to test whether the average of the difference is significantly different from μ0.

## Wilcoxon test

The Wilcoxon matched pairs test is a non-parametric equivalent of the paired t-test.

### Assumptions for Wilcoxon test:

• independent observations;
• interval scale, or ordinal scale with many alternatives.

Assumptions of paired t-test that do not apply to this test:

• Normal Distribution(s);
• no skew.

### Method for Wilcoxon test:

• Rank the differences (ignoring sign).
• Check which group is smaller (- or +).
• Sum ranks for this group: sum = test statistic T (strictly, after correcting for ties - where more than one observation share the same rank). (¿¿¿ or sum + and - ranks separately, and T is smaller of the two sums???).
• N = number of non-zero differences.
• if N<25, look up p in special tables (from N and T).
• if N≥25, a simple calculation using N & T gives a test statistic U - a standardised normal deviate, which is normally distributed, so p = 0.05 if U = 1.96, and so on.